A question about differential equations
Alasdair McAndrew <[email protected]> Sat, 17 Oct 2015 17:52:56 +1100
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In putting Axiom through its paces just recently (yes: Axiom, not a fork!),
I experimented with the ODE
y''+6y'+5y = 10x^2+4x+4exp(-x)
Now standard techniques (such as I teach my students), produce a solution
of the form
y = A*exp(-5x)+B*exp(-x)+2*x^2-4*x+4+x*exp(-x).
This is Axiom:
--(View this section in a fixed width font if it isn't shown as such)
(1) -> y:=operator 'y
(2) -> deq:=D(y(x),x,2)+6*D(y(x),x)+5*y(x)=10*x^2+4*x+4*exp(-x)
(3) -> sol:=solve(deq,y,x)
(3)
[
particular =
- x 6 2 - x 5 - 5x - x 2 - 5x
4x (%e ) + (10x - 16x + 16)(%e ) - %e %e - 2x %e
-----------------------------------------------------------------
- x 5
4(%e )
,
- x - 5x
basis= [%e ,%e ]]
Type: Union(Record(particular: Expression(Integer),basis:
List(Expression(Integer)))
--
of which the particular solution is a bit of a jumble. It doesn't seem to
be particularly simplifiable:
(4) -> simplifyExp(sol.particular)
(4) ->
2 - 5x - 6x
(8x - 16x + 16)%e + (4x - 1)%e
(4) ---------------------------------------
- 5x
4%e
--
Is there some way I can coerce Axiom into giving a particular solution to
such a straightforward ODE in a more simplified form?
Many thanks,
Alasdair
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<div dir=3D"ltr"><div>In putting Axiom through its paces just recently (yes=
: Axiom, not a fork!), I experimented with the ODE<br><br></div>y''=
+6y'+5y =3D 10x^2+4x+4exp(-x)<br clear=3D"all"><div><div><br></div><div=
>Now standard techniques (such as I teach my students), produce a solution =
of the form<br><br></div><div>y =3D A*exp(-5x)+B*exp(-x)+2*x^2-4*x+4+x*exp(=
-x).<br><br></div><div>This is Axiom:<br><br></div><div>--(View this sectio=
n in a fixed width font if it isn't shown as such)<br></div><div><br><s=
pan style=3D"font-family:monospace,monospace">(1) -> y:=3Doperator '=
y<br>(2) -> deq:=3DD(y(x),x,2)+6*D(y(x),x)+5*y(x)=3D10*x^2+4*x+4*exp(-x)=
<br>(3) -> sol:=3Dsolve(deq,y,x)<br>=C2=A0=C2=A0 (3)<br>=C2=A0=C2=A0 [<b=
r>=C2=A0=C2=A0 =C2=A0=C2=A0=C2=A0 particular =3D<br>=C2=A0=C2=A0 =C2=A0=C2=
=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 - x 6=C2=A0=C2=A0=C2=A0=C2=A0=
=C2=A0=C2=A0 2=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=
=C2=A0=C2=A0=C2=A0=C2=A0 - x 5=C2=A0=C2=A0=C2=A0=C2=A0 - 5x=C2=A0 - x=C2=A0=
=C2=A0=C2=A0=C2=A0 2=C2=A0 - 5x<br>=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 4x =
(%e=C2=A0=C2=A0 )=C2=A0 + (10x=C2=A0 - 16x + 16)(%e=C2=A0=C2=A0 )=C2=A0 - %=
e=C2=A0=C2=A0=C2=A0 %e=C2=A0=C2=A0=C2=A0 - 2x %e<br>=C2=A0=C2=A0=C2=A0=C2=
=A0=C2=A0=C2=A0 -----------------------------------------------------------=
------<br>=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=
=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=
=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=
=A0=C2=A0=C2=A0 - x 5<br>=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=
=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=
=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=
=C2=A0 4(%e=C2=A0=C2=A0 )<br>=C2=A0=C2=A0=C2=A0=C2=A0 ,<br>=C2=A0=C2=A0=C2=
=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 - x=C2=A0=
=C2=A0 - 5x<br>=C2=A0=C2=A0=C2=A0 basis=3D [%e=C2=A0=C2=A0 ,%e=C2=A0=C2=A0=
=C2=A0 ]]<br>Type: Union(Record(particular: Expression(Integer),basis: List=
(Expression(Integer)))</span><br><br>-- <br></div><div>of which the particu=
lar solution is a bit of a jumble.=C2=A0=C2=A0 It doesn't seem to be pa=
rticularly simplifiable:<br><br></div><div><span style=3D"font-family:monos=
pace,monospace">(4) -> simplifyExp(sol.particular)<br>(4) -> <br>=C2=
=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 2=C2=A0=C2=A0=C2=
=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 - 5x=C2=A0=
=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 - 6x<br>=
=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 (8x=C2=A0 - 16x + 16)%e=C2=A0=C2=
=A0=C2=A0=C2=A0 + (4x - 1)%e<br>=C2=A0=C2=A0 (4)=C2=A0 --------------------=
-------------------<br>=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=
=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=
=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 - 5x<br>=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=
=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=
=A0=C2=A0=C2=A0=C2=A0=C2=A0 4%e<br></span><br></div><div>-- <br></div><div>=
Is there some way I can coerce Axiom into giving a particular solution to s=
uch a straightforward ODE in a more simplified form?<br><br></div><div>Many=
thanks,<br></div><div>Alasdair<br><br><br></div><div><div class=3D"gmail_s=
ignature"><div dir=3D"ltr"><a href=3D"http://www.facebook.com/alasdair.mcan=
drew" target=3D"_blank"><img alt=3D"http://www.facebook.com/alasdair.mcandr=
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ng"></a> <a href=3D"http://numbersandshapes.net" target=3D"_blank"><img alt=
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/2796170/wordpress.png"></a><br></div></div>
</div></div></div>
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